Lili Lei

Empirical Localization of Observation Impact in Ensemble Kalman Filters

AbstractLocalization is a method for reducing the impact of sampling errors in ensemble Kalman filters. Here, the regression coefficient, or gain, relating ensemble increments for observed quantity y to increments for state variable x is multiplied by a real number α defined as a localization. Localization of the impact of observations on model state variables is required for good performance when applying ensemble data assimilation to large atmospheric and oceanic problems. Localization also improves performance in idealized low-order ensemble assimilation applications. An algorithm that computes localization from the output of an ensemble observing system simulation experiment (OSSE) is described. The algorithm produces localizations for sets of pairs of observations and state variables: for instance, all state variables that are between 300- and 400-km horizontal distance from an observation. The algorithm is applied in a low-order model to produce localizations from the output of an OSSE and the compu...

Comparisons of Empirical Localization Techniques for Serial Ensemble Kalman Filters in a Simple Atmospheric General Circulation Model

AbstractTwo techniques for estimating good localization functions for serial ensemble Kalman filters are compared in observing system simulation experiments (OSSEs) conducted with the dynamical core of an atmospheric general circulation model. The first technique, the global group filter (GGF), minimizes the root-mean-square (RMS) difference between the estimated regression coefficients using a hierarchical ensemble filter. The second, the empirical localization function (ELF), minimizes the RMS difference between the true values of the state variables and the posterior ensemble mean. Both techniques provide an estimate of the localization function for an observation’s impact on a state variable with few a priori assumptions about the localization function. The ELF localizations can have values larger than 1.0 at small distances, indicating that this technique addresses localization but also can correct the prior ensemble spread in the same way as a variance inflation when needed. OSSEs using ELF localiza...

Empirical Localization of Observations for Serial Ensemble Kalman Filter Data Assimilation in an Atmospheric General Circulation Model

AbstractThe empirical localization algorithm described here uses the output from an observing system simulation experiment (OSSE) and constructs localization functions that minimize the root-mean-square (RMS) difference between the truth and the posterior ensemble mean for state variables. This algorithm can automatically provide an estimate of the localization function and does not require empirical tuning of the localization scale. It can compute an appropriate localization function for any potential observation type and kind of state variable. The empirical localization algorithm is investigated in the Community Atmosphere Model, version 5 (CAM5). The empirical localization function (ELF) is computed for the horizontal and vertical separately so that the vertical localization is explored explicitly. The horizontal and vertical ELFs are also computed for different geographic regions. The ELFs varying with region have advantages over the single global ELF in the horizontal and vertical, because different...

A hybrid nudging-ensemble Kalman filter approach to data assimilation. Part II: application in a shallow-water model

ABSTRACT A hybrid nudging-ensemble Kalman filter (HNEnKF) data assimilation approach, explored in the Lorenz three-variable system in Part I, is tested in a two-dimensional shallow-water model for dynamic analysis and numerical weather prediction. The HNEnKF effectively combines the advantages of the ensemble Kalman filter (EnKF) and the observation nudging to achieve more gradual and continuous data assimilation by computing the nudging coefficients from the flow-dependent, time-varying error covariances of the EnKF. It can also transform the gain matrix of the EnKF into additional terms in the models predictive equations to assist the data assimilation process. The HNEnKF is tested for both a wave case and a vortex case with different observation frequencies and observation networks. The HNEnKF generally produces smaller root mean square (RMS) errors than either nudging or EnKF alone. It also has better temporal smoothness than the EnKF and lagged ensemble Kalman smoother (EnKS). The HNEnKF allows the gain matrix of the EnKF to be applied gradually in time, reducing the error spikes commonly found around the analysis times when using intermittent data assimilation methods. Therefore, the HNEnKF produces a seamless analysis with better inter-variable consistency and dynamic balance than the intermittent EnKF.

Model Space Localization Is Not Always Better Than Observation Space Localization for Assimilation of Satellite Radiances

AbstractCovariance localization is an essential component of ensemble-based data assimilation systems for large geophysical applications with limited ensemble sizes. For integral observations like the satellite radiances, where the concepts of location or vertical distance are not well defined, vertical localization in observation space is not as straightforward as in model space. The detailed differences between model space and observation space localizations are examined using a real radiance observation. Counterintuitive analysis increments can be obtained with model space localization; the magnitude of the increment can increase and the increment can change sign when the localization scale decreases. This occurs when there are negative background-error covariances and a predominately positive forward operator. Too narrow model space localization can neglect the negative background-error covariances and result in the counterintuitive analysis increments. An idealized 1D model with integral observations...

A hybrid nudging-ensemble Kalman filter approach to data assimilation. Part I: Application in the Lorenz system

ABSTRACT A hybrid data assimilation approach combining nudging and the ensemble Kalman filter (EnKF) for dynamic analysis and numerical weather prediction is explored here using the non-linear Lorenz three-variable model system with the goal of a smooth, continuous and accurate data assimilation. The hybrid nudging-EnKF (HNEnKF) computes the hybrid nudging coefficients from the flow-dependent, time-varying error covariance matrix from the EnKFs ensemble forecasts. It extends the standard diagonal nudging terms to additional off-diagonal statistical correlation terms for greater inter-variable influence of the innovations in the models predictive equations to assist in the data assimilation process. The HNEnKF promotes a better fit of an analysis to data compared to that achieved by either nudging or incremental analysis update (IAU). When model error is introduced, it produces similar or better root mean square errors compared to the EnKF while minimising the error spikes/discontinuities created by the intermittent EnKF. It provides a continuous data assimilation with better inter-variable consistency and improved temporal smoothness than that of the EnKF. Data assimilation experiments are also compared to the ensemble Kalman smoother (EnKS). The HNEnKF has similar or better temporal smoothness than that of the EnKS, and with much smaller central processing unit (CPU) time and data storage requirements.

Empirical Localization Functions for Ensemble Kalman Filter Data Assimilation in Regions with and without Precipitation

AbstractFor ensemble-based data assimilation, localization is used to limit the impact of observations on physically distant state variables to reduce spurious error correlations caused by limited ensemble size. Traditionally, the localization value applied is spatially homogeneous. Yet there are potentially larger errors and different covariance length scales in precipitation systems, and that may justify the use of different localization functions for precipitating and nonprecipitating regions. Here this is examined using empirical localization functions (ELFs). Using output from an ensemble observing system simulation experiment (OSSE), ELFs provide estimates of horizontal and vertical localization for different observation types in regions with and without precipitation. For temperature and u- and υ-wind observations, the ELFs for precipitating regions are shown to have smaller horizontal localization scales than for nonprecipitating regions. However, the ELFs for precipitating regions generally have ...

Multivariate Ensemble Sensitivity with Localization

AbstractEnsemble sensitivities have proven a useful alternative to adjoint sensitivities for large-scale dynamics, but their performance in multiscale flows has not been thoroughly examined. When computing sensitivities, the analysis covariance is usually approximated with the corresponding diagonal matrix, leading to a simple univariate regression problem rather than a more general multivariate regression problem. Sensitivity estimates are affected by sampling error arising from a finite ensemble and can lead to an overestimated response to an analysis perturbation. When forecasts depend on many details of an analysis, it is reasonable to expect that the diagonal approximation is too severe. Because spurious covariances are more likely when correlations are weak, computing the sensitivity with a multivariate regression that retains the full analysis covariance may increase the need for sampling error mitigation. The purpose of this work is to clarify the effects of the diagonal approximation, and investi...

A Four-Dimensional Incremental Analysis Update for the Ensemble Kalman Filter

AbstractThe analysis produced by the ensemble Kalman filter (EnKF) may be dynamically inconsistent and contain unbalanced gravity waves that are absent in the real atmosphere. These imbalances can be exacerbated by covariance localization and inflation. One strategy to combat the imbalance in the analyses is the incremental analysis update (IAU), which uses the dynamic model to distribute the analyses increments over a time window. The IAU has been widely used in atmospheric and oceanic applications. However, the analysis increment that is gradually introduced during a model integration is often computed once and assumed to be constant for an assimilation window, which can be seen as a three-dimensional IAU (3DIAU). Thus, the propagation of the analysis increment in the assimilation window is neglected, yet this propagation may be important, especially for moving weather systems.To take into account the propagation of the analysis increment during an assimilation window, a four-dimensional IAU (4DIAU) use...

Impacts of Frequent Assimilation of Surface Pressure Observations on Atmospheric Analyses

AbstractTo investigate the impacts of frequently assimilating only surface pressure (PS) observations, the Data Assimilation Research Testbed and the Community Atmosphere Model (DART/CAM) are used for observing system simulation experiments with the ensemble Kalman filter. An empirical localization function (ELF) is used to effectively spread the information from PS in the vertical. The ELF minimizes the root-mean-square difference between the truth and the posterior ensemble mean for state variables. The temporal frequency of the observations is increased from 6 to 3 h, and then 1 h. By observing only PS, the uncertainty throughout the entire depth of the troposphere can be constrained. The analysis error over the entire depth of the troposphere, especially the middle troposphere, is reduced with increased assimilation frequency. The ELF is similar to the vertical localization function used in the Twentieth-Century Reanalysis (20CR); thus, it demonstrates that the current vertical localization in the 20C...